An Improved Sliding Mode Control Technique to Mitigate Mismatched Parameter Uncertainties of Three-Phase Voltage Source Inverters

This paper proposes a theoretical and experimental implementation of an advanced sliding mode control (SMC) for a three-phase voltage source inverter (VSI) to achieve robustness against the unknown uncertainties of an LC filter. The significant contributions of the proposed SMC are summarized as considerations for the matched and mismatched uncertainties of the LC filter, more relaxed norm-bounds, dynamic characterization of sliding surface, and unique stability analysis. Unlike the conventional SMC techniques with matched uncertainties, the mismatched uncertainties in the state matrix are taken for the design of the proposed SMC. Also, the relaxed norm-bound designed for matched and mismatched uncertainties allows a wide range of variations in the values of LC filter. The voltage tracking errors are significantly reduced and the total harmonic distortions (THDs) are highly suppressed by characterizing the sliding surface in terms of flexible linear matrix inequalities (LMIs). Next, the stability analysis and reachability conditions are given using the Lyapunov criterion. The authenticity of the proposed SMC method is proved by TMS320LF28335 DSP based experimental results with a prototype 1-kVA test-bed. The comparative experimental results and analysis for the proposed SMC scheme, the conventional SMC scheme, and the conventional PI−PI control scheme are presented under the load step change, unbalanced load, and non-linear load step change with the parameter uncertainties to demonstrate the excellent performance of the proposed controller such as fast transient response, small steady-state error, and low THD.


I. INTRODUCTION
Three-phase voltage source inverter (VSI) is the most significant dc-ac component of many advanced applications such as uninterruptable power supplies, renewable energy systems, and active harmonic filters [1]. These applications require constant frequency and constant amplitude with sinusoidal voltage and low high-order harmonic components [2], whereas the quality of the inverter output voltage is seriously affected by sudden load changes, unbalanced loads, and non-linear loads [3]. In order to rectify the distortions and frequency-related switching voltage harmonics in the output voltage, the LC filter is connected to the output terminals The associate editor coordinating the review of this manuscript and approving it for publication was Jun Hu . of the VSI [4]. However, the parameter uncertainties associated with the LC filter due to aging and changing operating conditions complicate the control design [5] and stability analysis of the VSI. Also, the changes in the capacitance and inductance of an LC filter deteriorate the sinusoidal output voltage. Therefore, over the past few decades, considerable researches [6]- [12] have been done on the voltage control techniques to achieve a fast dynamic response [6], robustness to parameter variations, zero tracking error, and low total harmonic distortion (THD) [7].
The conventional linear proportional-integral (PI) control [8] is easy to implement but its poor capability to tolerate the effects of the parameter variations may deteriorate the control performance. Also, in spite of the extensive gain tuning, it has degraded the transient performance due to the trade-off between maintaining the system stability over the whole operating range and acquiring the adequate dynamic response during the transients. The linear quadratic regulator (LQR) [9] is presented to compensate for the parameter uncertainties for the VSI, which tunes the gains by selecting the entries of the weighting matrix. However, choosing a good set of values is very time-consuming. The model predictive control (MPC) [10] utilizes a system model at each sampling time to predict the output voltage for each potential switching state and the selection of the switching state criterion is established by the cost function. However, the MPC [11] requires the knowledge of model parameters and does not provide the analysis on stability and robustness. The multivariable feedback linearization control (FLC) [12] considers the non-linearity of a system to achieve a low THD under a non-linear load, but it has a computational complexity. The deadbeat control [13] has also gained attention as it provides the fast transient response by placing all the closed-loop poles at zero and the tracking error settles to zero within few sampling times. However, it is very sensitive to model uncertainties, parameters mismatches, and measurement noise especially for high sampling frequencies. In [14], the adaptive control with compensating terms is used to reduce the computational burden for the VSI, but it has a risk of divergence if the controller gains are not selected properly.
Meanwhile, the sliding mode control (SMC) is a non-linear control method applied in the VSI [15] due to its good dynamic response, robustness, and effective regulation properties in a wide range of operating conditions. Also, the SMC based VSI requires the switching control law and dynamic characterization of the sliding surface to ensure efficient voltage regulation and minimize the steady-state errors under all operating conditions. In [16], a switching control law is utilized in the SMC to drive the VSI uncertainties onto the sliding surface and maintain its trajectory on the sliding surface for all subsequent times. However, the voltage tracking error is not significantly reduced in the transient and steady-state operations. In order to minimize the steady-state errors, the integral term is added in the SMC, i.e., integral SMC (ISMC) for the dc-dc converters [17] and the dc-ac VSI [18], but it does not completely eliminate the voltage tracking errors and high-order harmonics. Meanwhile, the multi-resonant SMC (MRSMC) [19] employs multiple resonant terms in the sliding surface to eliminate the voltage tracking errors in the grid-connected VSI applications, but the computation of the resonant terms requires intensive calculations. In [20], the rotating sliding surface based SMC shows a good tracking performance under different load conditions, but the addition of the fuzzy controller to control the time-varying slopes of the SMC increases its complexity. In [21], the bounded norms for the parameter uncertainties are very strict, which provide insufficient robustness against the parameter uncertainties. Moreover, the conventional SMC techniques do not provide the stabilizing control laws that can suppress the mismatched uncertainties (i.e., uncertainties in the state model that do not satisfy matching conditions).
To accomplish fast transient response, small steady-state error (SSE), insensitivity to mismatched parameter variations, and low THD under critical load conditions, this paper proposes an advanced voltage control law for a three-phase VSI with an LC filter based on sliding mode control (SMC). The main contributions of the proposed SMC are summarized as considerations for the matched and mismatched uncertainties of the LC filter, more relaxed norm-bounds, dynamic characterization of sliding surface, switching control law, and unique stability analysis. First, both the matched and mismatched uncertainties of an LC filter are taken into account in the design of the proposed SMC. Second, the robust SMC law over a wide range of variations in the values of an LC filter is achieved by setting the relaxed norm-bounds for these matched and mismatched uncertainties. Third, the SMC design algorithm along with the characterization of the sliding surface and the sufficient condition for the existence of linear sliding surface is presented in terms of linear matrix inequalities (LMIs). Fourth, the switching control law contains a feedforward term, which rejects the disturbances of the VSI. Also, the switching control law drives the VSI trajectories to the sliding surface, and the sliding mode current observer is utilized to estimate the load currents. Fifth, the Lyapunov criterion is utilized to evaluate the stability of the sliding mode dynamics and the reachability conditions of the proposed SMC. Finally, the effectiveness of the proposed sliding mode voltage controller is verified by the experiment results via a prototype 1-kVA VSI test-bed having TMS320LF28335 DSP. The comparative and quantitative experimental results for the proposed SMC scheme, the conventional SMC scheme, and the conventional PI−PI control scheme are presented to prove fast transient response, small steady-state error (SSE), and low THD under load step change, unbalanced load, and non-linear load step change with the parameter uncertainties.

II. DYNAMIC MODEL OF A THREE-PHASE INVERTER
This section describes a dynamic model of the three-phase voltage source inverter (VSI) along with the mismatched parameter uncertainties. Fig. 1 shows the circuit topology of a three-phase inverter with an LC filter which consists of a dc-link voltage (V dc ), a three-phase pulse-width modulation (PWM) inverter (S 1 to S 6 ), an output LC filter (L f and C f ),  and a three-phase load. It is noted that the LC filter is used to provide sinusoidal output voltages to the load by attenuating the switching frequency components of the VSI. The proposed SMC of the VSI is designed in the dq-axis synchronous reference frame in which the three-phase AC signals are transformed to the DC or quasi-DC signals by applying the Park's transformation, which can efficiently control the load voltages under very harmonic polluted load currents. The dynamic model of the three-phase inverter in the dq-axis synchronous reference frame can be represented below [22]: (1) where v id , v iq , i id , and i iq are the dq-axis inverter voltages and currents, v Ld , v Lq , i Ld , and i Lq are the dq-axis load voltages and currents, ω is the angular frequency (ω = 2π · f ), f is the fundamental frequency of output voltage or current, k 1 = 1/C f , k 2 = 1/L f , and k 1 and k 2 are the parameter deviations in k 1 and k 2 due to aging and changing operating conditions [5], respectively.

III. SWITCHING CONTROL LAW DESIGN AGAINST MISMATCHED NORM−BOUNDED LC FILTER UNCERTAINTIES
This section presents the sliding surface design and the switching control law design for the proposed SMC. First, the proposed sliding surface is designed using a more relaxed norm-bounds for the matched and mismatched uncertainties of the LC filter. Moreover, the proposed sliding surface is dynamically characterized using the linear matrix inequality (LMI). Next, the switching control law is designed to drive the VSI trajectory to the switching surface and maintain a sliding mode condition.

A. SLIDING SURFACE DESIGN 1) RELAXED NORM-BOUNDS FOR MATCHED AND MISMATCHED UNCERTAINTIES
This subsection presents the relaxed norm-bounds for the matched and mismatched uncertainties of LC filter that are used to design the proposed sliding surface. According to the control theory, the matched uncertainties lie within the range space of the control input matrix, while the mismatched uncertainties do not lie within the range space of the input matrix. It should be noted that although the matched uncertainties can be easily removed by a properly designed control input, the mismatched uncertainties cannot be completely rejected [23]. Therefore, the design of a stabilizing controller for a mismatched uncertain system is a challenging task. Also, the robust SMC for the wide range variations in the parameter uncertainties of L f and C f is achieved by setting the relaxed conditions for the norm-bounds. The design of the mathematical norm-bounds is important so that the SMC can mitigate both matched and mismatched uncertainties.
Consequently, the parameter uncertainties k 1 and k 2 , due to aging and changing operating conditions [6], in (1) are norm-bounded with the assumptions that there exist the known finite constants ρ 1 , ρ 2 such that | k 2 | ≤ ρ 2 k 2 k 2 ; | k 1 | ≤ ρ 1 k 1 , and (ρ 1 / (1−ρ 1 )) k 1 . These inequalities imply that And for all the bounded but unknown disturbances with the bounded time derivative uncertainties, there exists a solution which is also uniformly bounded. Hence, by utilizing the relaxed norm-bounds which set very high limits for the parameter variations of L f and C f , the proposed SMC will produce the sinusoidal output voltage and ensure the robustness for all the wide range of parameter variations in the L f and C f within the limits set by the norm-bounds.

2) DYNAMIC CHARACTERIZATION OF SLIDING SURFACE
In this subsection, the new design of robust sliding surface against the matched and mismatched uncertainties is presented based on the LMI approach by setting the decay rate within the bounds. The reference values i d_ref and i q_ref for i id and i iq are explicitly explained as and v d_ref , v q_ref are the reference values of v Ld and v Lq . It is more convenient to use the error dynamics of the state variables for designing the sliding surface of the VSI as follows: The state-space model of the VSI employing the error dynamics is given aṡ where x represents the state variables, u R represents the reaching switching control law, A symbolizes the mismatched uncertainties, and d represents the disturbance inputs and matched parameter uncertainties, In d (i.e., d d and d q ), the first and second terms are the disturbance inputs, and the third and fourth terms denote the matched parameter uncertainties. Also, the mismatched uncertainties A are not required to be matched and they can be decomposed as follows: The parameter deviations in the LC filter are the norm bounded uncertainties (2) in the state matrix as well as the input matrix. The sufficient condition for the existence of the sliding surface sliding surface is presented in terms of LMIs by referring to [24]. Then, the sliding surface S is parameterized as where P = X −1 ∈ R 4×4 is a positive-definite matrix satisfying the following LMI condition In the above LMI condition, the symbol ∈ R 4×2 denotes the orthogonal complement of B and a full rank matrix such that T B = 0 and T = I . It should be noted that the sliding surface matrix S ∈ R 2×4 has a null space identical to B T P. In this regard, a transformation matrix T and its associated vector Z are defined as It is easy to show that T −1 = [X , B]. By using the Shur's complement lemma [25], we can show that the LMI condition (7) is equivalent to the following Riccati-like inequality It should be noted that the result of [25] implies the existence of some positive constant ρ M guaranteeing the feasibility of the LMI (7) or equivalently (9) for any 0 ≤ ρ 1 ≤ ρ M .
Remark 1: It is worth noting that the proposed sliding mode control (SMC) guarantees a better voltage control performance such as better robustness against the parameter variations of both L f and C f , more relaxed norm-bounds for the wide range variations of an LC filter, and more dynamic LMIs for the characterization of the sliding surface compared to the conventional SMC in [21]. First, the proposed SMC considers the parameter variations of both L f and C f , while [21] taking into consideration the parameter variations of L f only. If the parameter deviation k 2 of L f lies within the range space of the control input matrix B, it is taken as the matched uncertainties. On the other hand, if the parameter deviation k 1 of C f does not lie within the range space of B, it is considered as the mismatched uncertainties. Hence, unlike [21], both the matched and mismatched uncertainties (i.e., k 1 and k 2 ) are taken into account to design the proposed SMC law with low THD and small steady-state error in the output voltage. Second, unlike [21], the norm-bounded uncertainties presented in this paper are more relaxed (i.e., the parameters (e.g., L f and C f ) constrained by the norm-bounds can have a wide range of variations), which enable the robust behavior of the proposed SMC against the wide range uncertainties in L f and C f . Last, the proposed SMC employs the LMIs in the form of Riccati like inequalities which are more dynamic with a completely different structure from [21] to characterize the sliding surface, whereas [21] utilizes the Lyapunov inequality to design the sliding surface of the conventional SMC. Consequently, the proposed SMC method can achieve more robustness against the parameter variations, faster dynamic response, and smaller steady-state error in comparison to [21].

B. SWITCHING CONTROL LAW DESIGN
The objective of this subsection is to determine the reaching control law and the switched feedback gains which can drive the VSI trajectory to the switching surface and maintain a sliding mode condition. The variable structure control u R is designed such that the error dynamics x of the VSI outside the switching surface are driven to reach the sliding surface in finite time. The control variable u R = [v id v iq ] T can be divided into the following two terms where v ff = [v ffd v ffq ] T are the feedforward control terms for the compensation of matched disturbance inputs in d given as and v fb = [v fbd v fbq ] T are the switching feedback control terms to force the system trajectories to the sliding surface (σ = 0), presented as where SA = [D T d D T q ] T are the matrix product of the sliding surface matrix S and the control state matrix A, and Fig. 2 presents the block diagram of the proposed sliding mode control for a VSI with S and SA explained in (6) and (12), respectively. In this figure, the small values of ρ(t), i.e., ρ d and ρ q are more suitable for the steady-state operating condition. However, in terms of control activities, too small values of ρ d and ρ q can lead to slow transient responses. Therefore, the values of ρ d and ρ q should satisfy the two state sides, i.e., the steady-state and transient responses.

IV. SLIDING-MODE OBSERVER BASED CONTROL LAW AND STABILITY ANALYSIS
This section presents the sliding mode observer (SMO) that estimates the load currents i Ld and i Lq . Furthermore, the estimated load currents are used by the proposed SMC to design the SMO-based control law. Next, the stability analysis of the proposed SMO-based control law is presented using the Lyapunov function.

A. SLIDING-MODE LOAD CURRENT OBSERVER-BASED CONTROL LAW
To accurately estimate the load currents i Ld and i Lq under the wide range of parameter variations and noisy environments [25], the sliding mode observer (SMO) is employed due to its robustness to the external disturbances. Moreover, the load current SMO omits the situations of sudden breakdown of the current sensors and provides the reliability and cost reduction to the VSI. In this paper, it is assumed that the load currents change slowly during a small sampling period [25]. The sliding mode based load current observer for the i Ld and i Lq is represented as; where l d 1 and l q 1. By referring to [24], [26], [27] and references therein, the error dynamics of the SMO can be derived from (1) To estimate the load currents i Ld and i Lq [25], an averaging filter, i.e., a low pass filter is designed such that the characteristics of the signals are determined by the low frequencies and the unmodeled dynamics produces the high frequencies.
The estimated load currents can be calculated as; where 0 < τ d 1, 0 < τ q 1 are small filter time constants and s is the Laplace variable.
These estimated load currentsî Ld ,î Lq are utilized for constructing the error dynamics of the estimated state vector The SMO-based control law can be rewritten as; whereρ d andρ q are given bŷ Thus, the performance of the VSI will be robust to the measurement noise by utilizing the load currents provided by the SMO.

B. STABILITY ANALYSIS
This subsection discusses the stability of the SMO-based control law by analyzing the sliding mode dynamics and reachability conditions. The estimated state vectorx = [v de , v qe ,î de ,î qe ] T converges to the actual state x, i.e., the SMO-based control law (19) converges to (10). By the separation property given in [24], [26], [27], it is sufficient to show that the sliding mode dynamics with the switching control law (10) are stable and the reachability condition σ Tσ < 0 is guaranteed for all σ = 0. First, the stability analysis of the sliding mode dynamics is presented using the Lyapunov function. In this regard, the resulting reduced-order sliding mode dynamics restricted to the switching surface S with the input control law are defined using (8) as follows: where z ∈ R 2 . Then, from (5) the sliding mode dynamics are given by the following uncertain second-order equatioṅ The stability of the sliding mode dynamics requires the selection of a generalized Lyapunov function V(t,z) which is positive definite and has a negative time derivative in the region of attraction. Therefore, the Lyapunov function to yield an appropriate sliding mode for mismatched uncertainties is defined as And its derivative is calculated aṡ where the Riccati-like inequality (9) is used to prove the stability. The time derivative of the above Lyapunov function (24) is negative for all z = 0, which proves the stability of the sliding mode dynamics (22). Second, the stability analysis of the reachability condition is presented by highlighting that the sliding motion is reached with the switching feedback control law. The reachability condition is proven for all σ = 0 such that Hence, the convergence of σ is proven by the Lyapunov stability criteria. Thus, the reachability conditions are guaranteed, and the state-space model (5) with a SMC law is asymptotically stable in spite of the mismatched uncertainties and external disturbances. The physical meaning of the stability analysis and reachability conditions is that the proposed SMC will converge to the solution under all operating conditions. Then, the proof of the stability analysis of the sliding mode dynamics and reachability conditions provides the analytical analysis on a much wider scale compared to the numerical analysis.
Accordingly, the design procedure for the practical implementation of the proposed sliding mode voltage controller can be summarized by the following: Step 1) Build the system model (1) in d-q coordinate frame, and then derive the error dynamics (4) by using system parameters.
Step 2) Get the estimated load currentsî Ld ,î Lq from (17) using the observer gain parameters τ d , τ q , l d , and l q .
Step 3) Determine the solution X using the LMI conditions (7) and (9).
Step 4) Solve the sliding surface parameter matrix σ in (6) using the solution X .
Step 5) Calculate the switching feedback control law (19) using the controller gains ρ d and ρ q from (20). As depicted in Fig. 2, the proposed SMC can be easily implemented by solving S and SA. It is also noted that the tuning of the controller gains ρ d and ρ q is simple, so the proposed SMC does not require any exhaustive gain tuning as the gains are calculated from (20).

V. IMPLEMENTATION AND COMPARATIVE PERFORMANCE ANALYSIS UNDER WIDE-RANGE LC FILTER UNCERTAINTIES A. EXPERIMENTAL SETUP
To prove the authenticity and effectiveness of the proposed SMC algorithm, experiments with a prototype 1 kVA testbed are conducted for the following three scenarios, i.e., a load step change from no load to full load, an unbalanced load by suddenly opening one phase, and a non-linear load step change with a full-bridge diode rectifier. Note that these operating scenarios are selected to highlight the robustness of the proposed SMO under parameter variations. In this paper, the conventional SMC scheme [15] is used to carry out the comparative studies with the proposed SMC method. In addition, the conventional PI-PI control scheme is chosen for another comparative study to authenticate the superior performance of the proposed observer-based SMC scheme. Note that the PI gains are tuned using a general tuning rule mentioned in [3]. The optimum values of the control parameters for the proposed SMC are calculated as τ d = 0.0001, τ q = 0.0001, l d = 60000, l q = 50000, κ d = 0.1, and κ q = 0.1 due to the trade-off between the higher values increasing the chattering problem and the lower values reducing the convergence rate. The filter time constants τ d and τ q should be sufficiently small to preserve the control input component undistorted and large enough to eliminate the high-frequency components [28]. Meanwhile, the small values of the l d and l q can produce a slow convergence of the estimated load current, whereas the large values of the l d and l q can induce the chattering problem in the estimated load current values.
Furthermore, the control parameters κ d and κ q are the small offset values added in the gains of the SMC control to avoid the zero multiplication condition and execute the experiments. By solving the LMI condition (7), the solution matrix X is obtained and the sliding surface parameter matrix S is derived from (6) as follows:  flowchart of the DSP code to implement the experiments of the proposed SMC, respectively.
As shown in this figure, the load for each operating scenario is selected by the switch. The measured inverter phase currents (i i ) and line-to-neutral load voltages (v L ) in the abc reference frame are transformed into the corresponding quantities in the synchronously rotating d-q reference frame. Then, these dq-axis quantities are provided to the proposed sliding mode voltage controller. The command signals (v id and v iq ) are transformed into the quantities (v α and v β ) in the stationary α-β reference frame to implement a space-vector PWM (SVPWM) technique [29], [30].
In the experiments, the THD, root-mean square (RMS) values, and some waveforms for the experimental results are obtained using a Tektronix digital oscilloscope TDS5000, which is equipped with integrated power measurement and analysis (power analyzer) software. Note that the parameters of the LC filter are selected by considering several key factors such as cost, volume, and switching frequency. In the paper, the switching frequency (f s ) is selected as 5 kHz by taking into account a trade-off between the control performance and the efficiency with low switching loss. Even if numerous options of the L-C combinations are available for the selected f s , the capacitance C f = 7 µF is first chosen to minimize the no-load currents in capacitors and capacitor hardware cost, and then the inductance L f = 10 mH is properly selected based on the C f . In practical applications, the most common tolerance variations of the filter inductance (L f ) and filter capacitance (C f ) used as an output filter are within ±10%. To further justify the robustness against parameter variations, ±50% variations in respective L f and C f are assumed under all load conditions such as load step change, unbalanced load, and non-linear load step change. It is noted that the parameters of the non-linear load (i.e., R load = 200 and C load = 330 µF) shown in Fig. 3(a) are selected to achieve the maximum crest factor by considering the overcurrent limitation of our VSI. Table 1 presents the system parameters used to perform the experiments. In the following subsections, the detailed comparative studies with the experimental results will be presented.    In Fig. 4(a), the load output voltages are restored within less than only half a cycle after the load is suddenly applied. Even though the v L slightly exhibits the undershoot due to the underdamped nature of the control scheme, its rms magnitude deviates to less than 1% of the nominal value. In this scenario, the output voltage undershoot is less than 30.4% and the settling time (t s ) is about 1.2 ms. However, it is noted that the conventional SMC in Fig. 4(b) has a higher voltage undershoot of 55% and longer t s of 2.4 ms compared to the proposed SMC in Fig. 4(a). It can be seen in Fig. 4(c) that the conventional PI-PI control scheme has a higher voltage undershoot of 64.5% and longer t s of 2.85 ms compared to the proposed SMC. Next, the THDs for the proposed SMC in Fig. 4(a), the conventional SMC in Fig. 4(b), and the conventional PI−PI control scheme in Fig. 4(c) are 0.81%, 1.2%, and 1.53%, respectively. Hence, the low THD, small steady-state error (SSE), and robustness against parameter variations verify the better performance of the proposed SMC scheme compared to the conventional SMC scheme and the conventional PI−PI control scheme. Next, the behavior of the proposed SMC in Fig. 5(a) is compared to that of the conventional SMC in Fig. 5(b) under the scenario of an unbalanced load with 50% L f 0 and 150% C f 0 . In this scenario, phase A is suddenly opened to observe the  dynamic and steady-state behaviors of the proposed SMC and the conventional SMC. The waveforms of σ 1 and σ 2 verify that the proposed SMC can efficiently regulate the voltage under an unbalanced load condition due to the robust switching control law. Regardless of phase A opening, all output voltage waveforms in Fig. 5(a) are more sinusoidal than those in Fig. 5(b) because it is observed that the THDs/voltage dips in v L for the proposed SMC and the conventional SMC are 0.85%/2% and 1.31%/4%, respectively. Lastly, Fig. 6 illustrates the comparative performance analysis of the three control schemes (i.e., proposed SMC, conventional SMC, conventional PI−PI) under the scenario of the sudden non-linear load step change (i.e., a threephase diode rectifier shown in Fig. 3) with the 150% L f 0 and 150% C f 0 . It is noted that the non-linear load has the load resistor R load while the capacitor C load is pre-charged by the inverter. Because of the pre-charged capacitor voltage, the load current gets out of control for first one to two cycles during the transient-state after the insertion of the load resistor [3]. Fig. 6(a) highlights that the proposed SMC has a reduced THD (1.82%), faster t s (1.61 ms), smaller output voltage undershoots (35.6%), and smaller SSE (1.5 V). Meanwhile, the experimental results of the conventional SMC (THD: 3.38%; t s : 3.5 ms; voltage undershoot: 67.1%; SSE: 2.4 V) in Fig. 6(b) and the conventional PI-PI (THD: 4.31 %; t s : 4.8 ms; voltage undershoot: 71.9%; SSE: 2.7 V) in Fig. 6(c) display higher THD, longer t s , larger undershoot, and larger SSE compared to the proposed SMC. In addition, Fig. 6(a) shows that the output voltage is not almost affected despite the very high rate of change of the i L . The sliding surfaces σ 1 and σ 2 under this scenario show small overshoots which are reflected in the v L . Meanwhile, the higher perturbations in the v L shown in Fig. 6(b) and (c) prove that the conventional SMC and conventional PI−PI control schemes are not effective to mitigate the wide range variations of an LC filter under a sudden non-linear load step change.
Next, Fig. 7 presents the frequency spectrum analysis of the load voltage in volts as well as dBs for the proposed SMC, conventional SMC, and conventional PI−PI, respectively. As depicted in Fig. 7(a), the magnitudes of 5 th (1.5 V) and 7 th (0.8 V) harmonic components for the proposed SMC are significantly reduced due to the relaxed conditions designed for the norm-bounds of the proposed SMC. The 5 th (3.99 V) and 7 th (1.92 V) harmonic components in Fig. 7(b) for the conventional SMC and 5 th (4.89 V) and 7 th (3.28 V) harmonic components in Fig. 7(c) for the conventional PI−PI are higher compared to the proposed SMC.
From the experimental results of the three scenarios, the proposed SMC method provided the better performance such as good voltage regulation and lower THDs compared to the conventional SMC method. Moreover, the proposed SMC showed more robustness against the wide range parameter uncertainties of L f and C f . Last, Table 2 presents a summary of the comparative performance analysis of the proposed SMC, conventional SMC, and conventional PI−PI under all three operating scenarios described previously.

VI. CONCLUSION
This paper presents an advanced sliding mode voltage controller to mitigate the matched and mismatched uncertainties of the LC filter in a three-phase voltage source inverter. The sliding surface is designed using the LMIs which can select the relaxed norm-bounded conditions for the matched and mismatched uncertainties. The sliding mode observer, due to its robustness to external disturbances, is also employed to estimate the load currents. The stability of the sliding mode dynamics and the reachability conditions are presented using the Lyapunov theory. The more robustness against LC filter uncertainties, lower THD, smaller steady-state error, and faster transient response of the proposed SMC were verified under the three scenarios of a sudden load change, an unbalanced load, and a sudden non-linear load change compared to the conventional SMC and conventional PI−PI control. The proposed SMC in this paper can be applied to various applications such as active power filter, STATCOM, and ac motor drive to compensate for matched and mismatched uncertainties.